Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Mini Lesson Question

Question #1: Simplify Expressions Using Properties of Exponents

Which expression is correct?

$\frac {x^8}{x^4}=x^2$
$\frac {y^6}{y^2}=y^3$
$x^2 \cdot x^3=x^6$
$y \cdot y^3=y^4$

Simplify Expressions Using Properties of Exponents

In this unit, remember to double click on mathematical expressions/equations to enlarge, if needed.

In the expression $y^{4} \cdot y^{2}$ :

$y^{4} = y \cdot y \cdot y \cdot y$

The exponent 4 means there are 4 factors of $y$ .

$y^{2} = y \cdot y$

The exponent 2 means there are 2 factors of $y$ .

$y^{4} \cdot y^{2} = y \cdot y \cdot y \cdot y \cdot y \cdot y$

There are 6 factors of $y$ , which can be written as $y^{6}$ .

$y^{4} \cdot y^{2} = y^{6}$

Notice the exponent is the sum of the exponents of the factors.

When you multiply exponential expressions with the same base, you keep the base and add the exponents.

Product Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

$a^{m} \cdot a^{n} = a^{m + n}$

To multiply with like bases, add the exponents.

In the expression $\frac{y^{5}}{y^{3}}$ :

$y^{5} = y \cdot y \cdot y \cdot y \cdot y$

The exponent 5 means there are 5 factors of $y$ .

$y^{3} = y \cdot y \cdot y$

The exponent 3 means there are 3 factors of $y$ .

$\frac{y^{5}}{y^{3}} = \frac{y \cdot y \cdot y \cdot y \cdot y}{y \cdot y \cdot y} = y \cdot y$

There are 2 factors of $y$ , which can be written as $y^{2}$ .

$\frac{y^{5}}{y^{3}} = y^{2}$

Notice the exponent is the difference of the exponents of the factors.

When you divide exponential expressions with the same base, you keep the base and subtract the exponents.

Quotient Property for Exponents

If $a$ is a real number, $a \neq 0$ , and $m$ and $n$ are integers, then

$\frac{a^{m}}{a^{n}} = a^{m - n}$ , $m > n$ and $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ , $n > m$

Here’s how to simplify expressions using properties of exponents.

1. $x^{2} \cdot x^{3}$

The bases are the same. To multiply, use the Product Property for Exponents.

$x^{2} \cdot x^{3} = x^{2 + 3} = x^{5}$ . To multiply with like bases, add the exponents.

2. $y \cdot y^{4}$

The bases are the same. To multiply, use the Product Property for Exponents.

Remember: A variable with no exponent shown has an exponent of $1$ , $y = y^{1}$ .

$y \cdot y^{4} = y^{1} \cdot y^{4} = y^{1 + 4} = y^{5}$ . To multiply with like bases, add the exponents.

3. $\frac{x^{5}}{x^{2}}$

The bases are the same. To divide, use the Quotient Property of Exponents.

$\frac{x^{5}}{x^{2}} = x^{5 - 2} = x^{3}$ . To divide with like bases, subtract the exponents.

4. $\frac{x^{4} y z}{x^{2} y^{2} z}$

The expression contains like bases. Use the Quotient Property of Exponents.

$\frac{x^{4} y z}{x^{2} y^{2} z} = \frac{x^{4} y^{1} z^{1}}{x^{2} y^{2} z^{1}} = \frac{x^{4 - 2} z^{1 - 1}}{y^{2 - 1}} = \frac{x^{2} z^{0}}{y^{1}} = \frac{x^{2}}{y}$ . To divide with like bases, subtract the exponents.

Remember: Any nonzero number to the zero power is 1.

Try it

Try It: Simplify Expressions Using Properties of Exponents

Simplify the expressions.

1. $x^{3} \cdot x^{4}$

Here is how to simplify expressions using properties of exponents.

$x^{3} \cdot x^{4} = x^{3 + 4} = x^{7}$ . To multiply with like bases, add the exponents.

2. $\frac{x^{9}}{x^{3}}$

Compare your answer:

$\frac{x^{9}}{x^{3}} = x^{9 - 3} = x^{6}$ . To divide with like bases, subtract the exponents.

3. $\frac{x^{4} y^{2} z^{5}}{x^{6} y z^{3}}$

Compare your answer:

$\frac{x^{4} y^{2} z^{5}}{x^{6} y z^{3}} = \frac{y^{2 - 1} z^{5 - 3}}{x^{6 - 4}} = \frac{y z^{2}}{x^{2}}$ . To divide with like bases, subtract the exponents.

Check Your Understanding

Which expression is equivalent to $y^{6}$ ?

Multiple Choice:

$y^{2} \cdot y^{3}$
$y \cdot y^{6}$
$\frac{y^{12}}{y^{2}}$
$\frac{y^{12}}{y^{6}}$

Take a moment to think about what you learned in the mini-lesson review.

Videos: Simplifying Expressions with Exponents

Khan Academy: Exponent properties involving products

Watch this video to see how to multiply expressions with exponents.

Access multimedia content

Khan Academy: Exponents properties with quotients

Watch this video to see how to divide expressions with exponents.